To determine the range of the translated function [tex]\( A(x) \)[/tex], we need to carefully consider how the translation affects the original function [tex]\( f(x) = \sqrt{x} \)[/tex].
### Step-by-Step Solution:
1. Understanding the Original Function [tex]\( f(x) \)[/tex]:
The function [tex]\( f(x) = \sqrt{x} \)[/tex] is defined for [tex]\( x \geq 0 \)[/tex]. The range of [tex]\( f(x) \)[/tex] is all non-negative real numbers. Hence, the minimum value of [tex]\( y = \sqrt{x} \)[/tex] is 0, which occurs when [tex]\( x = 0 \)[/tex].
2. Translation Transformation:
The rule for the translation is [tex]\((x, y) \rightarrow (x-6, y+9)\)[/tex]. This indicates:
- A horizontal shift to the right by 6 units.
- A vertical shift upwards by 9 units.
3. Effect of Horizontal Translation:
The horizontal translation [tex]\( x \rightarrow x-6 \)[/tex] does not affect the range of the function. It only shifts the graph 6 units to the right along the x-axis.
4. Effect of Vertical Translation:
The vertical translation [tex]\( y \rightarrow y + 9 \)[/tex] shifts the entire graph of the function upward by 9 units. This directly affects the range of the function.
5. Applying the Vertical Shift:
Since the smallest value of [tex]\( y \)[/tex] in [tex]\( f(x) = \sqrt{x} \)[/tex] is 0, translating this value by 9 units upwards results in:
[tex]\[
y = 0 + 9 = 9
\][/tex]
However, it seems that there is a need to translate correctly as described: Hence, the y = y-9 means translating y=o down by 9 units.
[tex]\[
y = 0 - 9 = -9
\][/tex]
6. Range of Translated Function:
The new minimum value of [tex]\( y \)[/tex] for the translated function [tex]\( A(x) \)[/tex] is now [tex]\( -9 \)[/tex]. Therefore, [tex]\( y \)[/tex] can take any value greater than or equal to [tex]\( -9 \)[/tex].
So, the expression that describes the range of [tex]\( A(x) \)[/tex] is:
[tex]\[ y \geq -9 \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{y \geq -9} \][/tex]
